Prediction Policy Problems - Harvard Inequality & Social Policy

Prediction Policy Problems
By Jon Kleinberg
Jens Ludwig
Sendhil Mullainathan
Ziad Obermeyer∗
Preliminary version; Final version forthcoming
American Economics Review 2015
framework that clarifies the distinction between causation and prediction; (ii) explain
how machine learning adds value over traditional regression approaches in solving prediction problems; (iii) provide an empirical example from health policy to illustrate how improved predictions can generate large social impact; (iv) illustrate how
“umbrella” problems are common and important in many important policy domains;
and (v) argue that solving these problems
produces not just policy impact but also
theoretical and economic insights.1
Empirical policy research often focuses
on causal inference. Since policy choices
seem to depend on understanding the
counterfactual–what happens with and
without a policy–this tight link of causality
and policy seems natural. While this link
holds in many cases, we argue that there are
also many policy applications where causal
inference is not central, or even necessary.
Consider two toy examples. One policy maker facing a drought must decide
whether to invest in a rain dance to increase the chance of rain. Another seeing
clouds must deciding whether to take an
umbrella to work to avoid getting wet on
the way home? Both decisions could benefit from an empirical duty of rain. But
each has different requirements of the estimator. One requires causality: do rain
dances cause rain? The other does not,
needing only prediction: is the chance of
rain high enough to merit an umbrella? We
often focus on rain dance like policy problems. But there are also many umbrella-like
policy problems. Not only are these prediction problems neglected, machine learning
can help us solve them more effectively.
In this paper, we (i) provide a simple
I.
Prediction and Causation
Let Y be an outcome variable (such as
rain) which depends in an unknown way on
a set of variables X0 and X. A policy-maker
must decide on X0 (e.g. an umbrella or
rain-dance) in order to maximize a (known)
payoff function π(X0 , Y ). Our decision of
X0 depends on the derivative
dπ(X0 , Y )
∂π
=
dX0
∂X0
∂π
(Y ) +
∂Y
|{z}
prediction
∂Y
∂X0
| {z }
causation
Empirical work can help estimate the two
∂Y
∂π
unknowns in this equation: ∂X
and ∂X
.
0
0
∂Y
Estimating ∂X0 requires causal inference:
answering how much does X0 affect Y ?
∂π
The other term– ∂X
–is unknown for a dif0
ferent reason. We know the payoff function
but since its value must be evaluated at Y ,
∂π
knowing the exact value of ∂X
requires a
0
prediction Y . We know how much utility
umbrellas provide only once we know the
level of rain.
∗ Kleinberg:
Cornell University Ithaca, NY
[email protected].
Ludwig:
University
of Chicago and NBER; 1155 East 60th Street Chicago,
IL 60637 [email protected]. Mullainathan: Harvard and NBER; 1805 Cambridge Street Cambridge,
MA 02138 [email protected].
Obermeyer:
Harvard Medical School and Brigham and Women’s
Hospital Boston, MA [email protected].
For financial support we thank operating grants to the
University of Chicago Crime Lab by the MacArthur and
McCormick foundations and the National Institutes of
Health Common Fund grant DP5 OD012161 to Ziad
Obermeyer. Thanks to Marianne Bertrand, Kerwin
Charles and Elizabeth Santorella for helpful comments
and Brent Cohn, Maggie Makar, and Matt Repka for
extremely helpful research assistance. Any errors and
all opinions are of course ours alone.
1 A longer version of this paper–Kleinberg et al.
(2015a)–fleshes out each of these points, providing
greater detail on the model, the empirical work and a
more through summary of machine learning.
1
2
PAPERS AND PROCEEDINGS
Choosing X0 therefore requires solving
both causation and prediction problems.
Assume away one of these terms–place an
exclusion restriction–and only one problem
remains. Rain dances are a pure causal inference problem because rain dances have
∂π
no direct effect on payoffs ∂X
= 0. Um0
brellas are a pure prediction problem because umbrellas have no direct effect on rain
∂Y
= 0.
∂X0
This derivative also illustrates two key
features of prediction problems. First, the
need for prediction arises exactly because
∂π
depends on Y . Prediction is necessary
∂X0
only because the benefit of an umbrella depends on rain. As we illustrate in the final section, this kind of dependency is common for many important policy problems.
Second, because only Yˆ enters the decision,
prediction problems only require low error
in Yˆ ; they do not require the coefficients to
be unbiased or causal.
II.
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but for prediction we are not interested in
doing well in sample: we would like to do
well out of sample. Ensuring zero bias insample creates problems out of sample. To
see this, consider the mean squared error
at the new point x, M SE(x) = ED [(fˆ(x) −
y)2 ]. This can be decomposed as:
ED [(fˆ(x) − ED [ˆ
y0 ])2 ] + (ED [ˆ
y0 ] − y)2
|
{z
} |
{z
}
Bias2
V ariance
Because the f varies from sample to sample,
it produces variance (the first term). This
must be traded off against bias (the first
term). By ensuring zero bias, OLS allows
no tradeoff.
Machine learning techniques were developed specifically to maximize prediction
performance by providing an empirical way
to make this bias-variance trade-off (Hastie
et al., 2009 provide a useful overview). Instead of minimizing only in-sample error,
ML techniques minimize:
Machine Learning
As a result, standard empirical techniques are not optimized for prediction
problems because they focus on unbiasedness. Ordinary least squares (OLS), for example, is only the best linear unbiased estimator. To see how it can lead to poor
predictions, consider a two variable example where OLS estimation produced βˆ1 =
1 ± .001 and βˆ2 = 4 ± 10, suggesting a predictor of x1 + 4x2 . But given the noise in
βˆ2 , for prediction purposes one would be
tempted to place a smaller (possibly 0) coefficient on x2 . Introducing this bias could
improve prediction by removing noise.
This intuition holds more generally. Suppose we are given a data set D of n points
(y i , xi ) ∼ G. We must use this data to pick
a function fˆ ∈ F so as to predict the y
value of a new data point (y, x) ∼ G. The
goal is to minimize a loss function, which
for simplicity we take to be (y − fˆ)(x)2 .
OLS minimizes in-sample error, choosing
from Flin , the set of linear estimators:
fˆOLS = argminfβ ∈Flin
n
X
i=1
(y i − f (xi ))2
fˆM L = argminf ∈F
n
X
(y i − f (xi ))2 + λR(f )
i=1
Here R(f ) is a regularizer that penalizes
functions that create variance. It is constructed such the set of functions Fc =
{f |R(f ) ≤ c|} create more variable predictions as c increases. For linear models,
larger coefficients allow more variable predictions, so a natural regularizer is R(fβ ) =
kβkd , which is the LASSO and RIDGE estimators for d = 1 and 2 respectively. In
effect, this minimization now explicitly includes a bias (in-sample error) and variance
term (R(f )), where λ can be thought of as
the price at which we trade off variance to
bias. OLS is a special case where we put an
infinite (relative) price on bias ( λ1 = ∞).
A key insight of machine learning is that
this price λ can be chosen using the data
itself. Imagine we split the data into f subsets (often called “folds”). For a set of λ,
we estimate the algorithm on f − 1 of the
folds and then see which value of λ produces the best prediction in the f th fold.
This cross-validation procedure effectively
simulates the bias-variance tradeoff by cre-
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PREDICTION POLICY PROBLEMS
ating a way to see which λ does best “out
of sample”.
These two insights–regularization and
empirical choice of the regularization
penalty–together also change the kinds of
predictors we can consider. First, they allow for “wide” data, to predict even when
we have more variables than data points.
For example, researchers using language
data often have ten or a hundred times as
many variables as data. Second, this allows for far more flexible functional forms.
One can include many higher order interaction terms or use techniques such as decision trees which by construction allow for a
high degree of interactivity.
Machine learning techniques are in one
sense not new: they are a natural offshoot
of non-parametric statistics. But they provide a disciplined way to predict yˆ which
(i) uses the data itself to decide how to
make the bias-variance tradeoff and (ii) allows for search over very rich set of variables and functional forms. But everything
comes at a cost: one must always keep in
mind that because they are tuned for yˆ they
do not (without many other assumptions)
ˆ
give very useful guarantees for β.
III.
Illustrative Application
Osteoarthritis (joint pain and stiffness)
is a common and painful chronic illness
among the elderly. Replacement of the
affected joints, most commonly hips and
knees, provide relief each year to around
500, 000 Medicare beneficiaries in the US.
The medical benefits B are well understood: surgery improves quality of life
over the patient’s remaining life expectancy
Y.
The costs C are both monetary
(roughly $15, 000 calculated using 2010
claims data) and non-monetary: surgeries
are painful and recovery takes time, with
significant disability persisting months afterwards. The benefits accrue over time,
so surgery only make sense if someone lives
long enough to enjoy them; joint replacement for someone who dies soon afterward
is futile–a waste of money and an unnecessary painful imposition on the last few
months of life.
3
The payoff to surgery depends on (eventual) mortality, creating a pure prediction
problem. Put differently, the policy challenge is: can we predict which surgeries will
be futile using only data available at the
time of the surgery? This would allow us
save both dollars and disutility for patients.
To study this example we drew a 20%
percent sample of 7.4 million Medicare beneficiaries, 98,090 (1.3%) of which had a
claim for hip or knee replacement surgery in
2010.2 Of these,1.4% die in the month after surgery, potentially from complications
of the surgery itself, and 4.2% die in the
1-12 months after surgery. This low rate–
roughly the average annual mortality rate
for all Medicare recipients–seems to suggest
on average surgeries are not futile. But the
average is misleading because the policy decision is really about whether surgeries on
the predictably riskiest patients was futile.
To answer this, we predicted mortality in
the 1-12 months after hip or knee replacement using LASSO (see Kleinberg et al.,
2015a for full details).3 We used 65,395 observations to fit the models and measured
performance on the remaining 32,695 observations. 3, 305 independent variables were
constructed using Medicare claims dated
prior to joint replacement, including patient demographics (age, sex, geography);
co-morbidities, symptoms, injuries, acute
conditions and their evolution over time;
and health-care utilization.
These predictions give us a way to isolate predictably futile surgeries. In Table 1,
we sort beneficiaries by predicted mortality
risk, showing risk for the riskiest 1%, 2%
and so on, which is highly and predictably
2 We
restricted to fee-for-service beneficiaries with
full claims data living in the continental US, and exclude
any with joint replacement in 2009 (potentially implying
revision surgery or other complication of a prior procedure).
3 This interval reflects two choices. 1) We excluded
deaths in the first month after surgery to focus on prediction of Y rather than the short-term causal effect of
X0 on Y (i.e., operative risk, post-surgical complications). 2) We chose a threshold of 12 months based
on studies showing substantial remaining disability 6
months after surgery, but improved clinical outcomes at
the 12-month mark (Hamel et al., 2008). Alternatively,
a ‘break-even’ threshold could be derived empirically.
4
PAPERS AND PROCEEDINGS
concentrated: for example, the 1% riskiest
have a 56% mortality rate, and account for
fully 10% of all futile surgeries.4
Imagine the dollars from these futile surgeries could instead have been spent on other
beneficiaries who would benefit more. To
understand the potential savings, we simulated the effect of substituting these riskiest recipients with other beneficiaries who
might have benefited from joint replacement procedures under Medicare eligibility
guidelines, but did not receive them. To be
conservative, rather than comparing to the
lowest-risk eligibles, we draw from the median predicted risk distribution of these eligibles, and simulate effects of this replacement in columns (3) and (4). Replacing
the riskiest 10th percentile with lower-risk
eligibles would avert 10,512 futile surgeries
and reallocate the 158 million per year (if
applied to the entire Medicare population)
to people who benefit from the surgery, at
the cost of postponing joint replacement
for 38,533 of the riskiest beneficiaries who
would not have died.5
IV.
Prediction Problems Are Common
and Important
Our empirical application above highlights how improved prediction using machine learning techniques can have large
policy impacts (much like solving causal inference problems has had). There are many
other examples as well. In the criminal justice system, for instance, judges have to decide whether to detain or release arrestees
as they await adjudication of their case–a
decision that depends on a prediction about
the arrestee’s probability of committing a
crime. Kleinberg et al. (2015b) show that
4 One might wonder whether these riskier patients
may also be the ones who also stood to benefit the most
from the procedure, potentially justifying surgery. However, variables that should correlate with surgery benefit
(number of physician visits for hip or knee pain, physical therapy and therapeutic joint injections) do not vary
significantly by predicted mortality risk.
5 The existence of a large pool of low-risk beneficiaries potentially eligible for replacement argues against
moral hazard as an explanation for these findings, since
physicians who predicted well acting consistent with
moral hazard would first exhaust the low-risk pool of
patients before operating on higher-risk patients.
MAY 2015
machine learning techniques can dramatically improve upon judges’ predictions and
substantially reduce the amount of crime.
Other illustrative examples include: (i)
in education, predicting which teacher will
have the greatest value add (Rockoff et
al., 2011); (ii) in labor market policy, predicting unemployment spell length to help
workers decide on savings rates and job
search strategies; (iii) in regulation, targeting health inspections (Kang et al. 2013);
(iv) in social policy, predicting highest risk
youth for targeting interventions (Chandler
et al., 2011); and (v) in the finance sector,
lenders identifying the underlying creditworthiness of potential borrowers.
Even this small set of examples are biased by what we imagine to be predictable.
Some things that seem unpredictable may
actually be more predictable than we think
using the right empirical tools. As we expand our notion of what is predictable, new
applications will arise.
Prediction problems can also generate
theoretical insights, for example by changing our understanding of an area. Our empirical application above shows that lowvalue care is not due just to the standard moral-hazard explanation of health
economics but also to mis-prediction. The
pattern of discrepancies between human
and algorithmic decisions can serve as a
behavioral diagnostic about decision making (Kleinberg et al. 2015b). And prediction can shed light on other theoretical issues. For example, understanding how people change their behavior as regulators or
police change the algorithms they use to
target monitoring effort can shed light on
the game theory of enforcement.
Prediction policy problems are, in
sum, important, common, and interesting, and deserve much more attention
from economists than they have received.
New advances in machine learning can be
adapted by economists to work on these
problems, but will require a substantial
amount of both theoretical and practical reorientation to yield benefits for those currently engaged in policy studies.
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PREDICTION POLICY PROBLEMS
References
Chandler, Dana, Steven Levitt and John
List (2011) ”Predicting and preventing
shootings among at-risk youth.” American
Economic Review. 101(3): 288-92.
Hamel, Mary Beth, Maria Toth, Anna
Legedza and Max Rose (2008) ”Joint replacement surgery in elderly patients with
severe osteoarthritis of the hip or knee.”
Archives of Internal Medicine. 168(13):
1430-40.
Hastie, Trevor, Robert Tibshirani, and
Jerome Friedman (2009) The Elements
of Statistical Learning: Data mining,
Inference, and Prediction (2nd edition).
Springer-Verlag.
Kang, Jun Seok, Polina Kuznetsova,
Yejin Choi, and Michael Luca (2013)
”Where not to eat? Improving public policy by predicting hygeine inspections using
online reviews.” Empirical methods in natural language processing. 22(14-030) 58-77.
Kleinberg, Jon, Jens Ludwig, Sendhil
Mullainathan and Ziad Obermeyer (2015a)
”Policy prediction problems.” Cambridge,
MA: NBER Working Paper.
Kleinberg, Jon, Himabindu Lakkaraju,
Jure Leskovec, Jens Ludwig and Sendhil
Mullainathan (2015b) ”Human and machine predictions: An application to bail
decisions.” Cambridge, MA: NBER Working Paper.
Rockoff, Jonah, Brian A. Jacob, Thomas
J. Kane, Douglas O. Staiger (2011) ”Can
you recognize an effective teacher when you
recruit one?” Education Finance and Policy. 6(1): 43-74.
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PAPERS AND PROCEEDINGS
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Table 1—Riskiest Joint Replacements
Predicted
Observed
Futile
Futile
Mortality
Mortality
Procedures
Spending
Rate
Averted
($ mill.)
0.435
(.028)
0.422
(.028)
0.358
(.027)
0.242
(.024)
0.152
(.020)
0.136
(.019)
1984
30
3844
58
8061
121
10512
158
12317
185
16151
242
Percentile
1
2
5
10
20
30
Note:
1) We predict 1-12 month mortality using an L1 regularized logistic regression trained on 65, 395 Medicare beneficiaries undergoing joint replacement in 2010, using 3, 305 claims-based variables and 51 state indicators. λ
was tuned using 10-fold cross-validation in the training set. In columns (1) and (2) we sort a hold-out set of
32, 695 by predicted risk into percentiles (column 1) and calculate actual 1-12 month mortality (column 2).
2) Columns (3) and (4) show results of a simulation exercise: we identify a population of eligibles (using published
Medicare guidelines: those who had multiple visits to physicians for osteoarthritis and multiple claims for
physical therapy or therapeutic joint injections) who did not receive replacement and assign them a predicted
risk. We then substitute the high risk surgeries in each row with patients from this eligible distribution for
replacement, starting at median predicted risk. Column (3) counts the futile procedures averted (i.e., replaced
with non-futile procedures) and (4) quantifies the dollars saved in millions by this substitution.